Among the games played by children which require dexterity in movement coupled with a sense of balance, one of the first which comes to mind is hop skotch. In this game, the player must not only be able to leap or jump forward, but also maintin his balance, be it on one foot in the early plays, but also be able to convert the play from one foot to two feet, back to one foot and then to two feet again. Being played on a flat surface, the requirement for balance is minimized to the extent of not losing the sense of vertical balance so at to have to take an additional step to avoid falling out of the particular space.
Heretofore, there are no known games which require a player to not only maintain his balance but also move himself forward toward a pre-established goal.
Also to be considered as a possible contrast, is the attraction of walking, or at least trying, along a train rail. This exercise in balance is fortunate in that a would be walker is only 3-4 inches from the ground. There is also the attraction of walking along the top rail of a wooden fence. This attraction is only negated by the possible injury resulting from a fall from the top rail. Such self-inducted games require only a sense of balance on a stationary horizontal member.
There is, however, a sport which requires ability and also agility, to maintain one's equilibrium on an object which has longitudinal stability but is also subject to rotational instability if not properly controlled. In short this is the log rolling contest which is a major part of all lumberjack contests. Such contest require a body of water in which a log is floating. This content requires an ability to not only balance oneself on the log but to do so while the log is being rotated either by one's self or the opponent.
There is no known game which requires a player to maintain his balance on a cylindrical object and at the same time move himself along a playing surface.